phase measurement & quantum algorithms dominic berry iqc university of waterloo howard wiseman...

Phase Measurement & Quantum Algorithms

Dominic BerryIQC University of Waterloo

Howard Wiseman Geoff PrydeBrendon HigginsGuoyong Xiang Griffith University

Steve Bartlett University of Sydney

Morgan Mitchell ICFO

Tim Ralph University of Queensland

Trevor WheatleyElanor Huntington UNSW

Hidehiro YonezawaDaisuke NakaneHajime AraoAkira Furusawa University of Tokyo

Damian Pope Perimeter Institute

Barry SandersAlex Lvovsky University of Calgary

Andrew Childs University of Waterloo

Jason TwamleyAlexei GilchristGavin BrennenRessa Said Macquarie University

Outline

1. Phase measurement

2. Anyon simulation

3. Photon processing

4. Quantum algorithms

5. Research plans

Core work of the Centre for Quantum Dynamics.

Phase measurement

Communication

Frequency and time measurement

Distance measurement

Phase measurement

Phase measurement

Multipass interferometry

Nonadaptive interferometry

Multiphoton interferometry

Tracking a fluctuating phase

Loss resistant states

Magnetometry

Interferometry

Simple inputs and measurements give Standard Quantum Limit:

est

N photons

(t )

Interferometry

N photons est

More advanced inputs and measurements give Heisenberg Limit:

(t )

Theoretical work with Howard Wiseman showed feedback can give this result (PRL, 2000).

NOON state interferometry

input state

,0 0,N N

est

/

p( )

B. C. Sanders, Phys. Rev. A 40, 2417 (1989).

Ambiguity problem due to multiple fringes.

,0 0,iNe N N

Multipass interferometry

Similar ambiguity problem.

1 photon (t)

est

/

p( )

Multipass interferometry

Resolving the ambiguity.

/

p( )

1 photon

est

(t)

Multipass interferometry

/

p( )

(t)1 photon

est

Resolving the ambiguity.

Multipass interferometry

/

p( )

1 photon

est

Resolving the ambiguity.

(t)

Experimental results

M = 6

SQL

M = 1

vari

ance

N

number of resources, N

theoretical limit

B. L. Higgins, DWB, S. D. Bartlett, H. M. Wiseman & G. J. Pryde, Nature 450, 393-396 (2007).

Nonadaptive interferometry

Previously it was expected that we can’t achieve the theoretical limit without adaptive measurements.

Not so! We can achieve the theoretical limit with just a sequence of nonadaptive measurements and multiple passes.

Not only this, we can prove that it is at the theoretical limit!

Experimental resultsB. L. Higgins, DWB, S. D. Bartlett, M. W. Mitchell, H. M. Wiseman & G. J. Pryde, New Journal of Physics 11, 073023 (2009).

number of resources, N

SQL

theoretical limit

nonadaptive

hybrid

stan

dard

dev

iati

on

N 1

/2

Multiphoton interferometry

Multiphoton interferometry

Multiphoton interferometry

Multiphoton interferometry

Use three different states.

Determine a sequence of states for a given total photon number N such that the final variance is minimised.

Use feedback such that the expected variance after the next detection is minimised.

Adaptive estimation with entanglement

HL

SQL

photon number, N

stan

dard

dev

iati

on

N 1

/2

G. Y. Xiang, B. L. Higgins, DWB, H. M. Wiseman & G. J. Pryde, Nature Photonics 5, 43-47 (2011).

Tracking a fluctuating phase

( )I t

DWB & H. M. Wiseman, Phys. Rev. A 65, 043803 (2002).

DWB & H. M. Wiseman, Phys. Rev. A 73, 063824 (2006).

Tracking a fluctuating phase

a) Signal and local oscillator generation.

b) Adaptive phase estimation.

c) Dual homodyne phase estimation.

LO = local oscillator;RF = radio-frequency;EOM = electro-optic

modulator;WGM = waveguide modulator; LPF = low-pass filter;MCC = mode-cleaning cavity;AOM = acousto-optic

modulator.

Tracking a fluctuating phase

dual homodyne filtered

T. A. Wheatley, DWB, H. Yonezawa, D. Nakane, H. Arao,D. T. Pope, T. C. Ralph, H. M. Wiseman, A. Furusawa &

E. H. Huntington, Physical Review Letters 104, 093601 (2010).

Loss resistant states NOON states are

very sensitive to loss.

States with optimal loss resistance are difficult to produce.

I am working on simpler methods to produce near-optimal states.

output

NOON

best from beam splitter

optimal loss tolerant

near-optimal states

coherent states

Magnetometry Advances in nitrogen-

vacancy centres offer ability to map magnetic fields at nanoscale resolution.

With longer T2 times, the measurements have a similar problem with ambiguity.

We can apply methods from optical measurements to obtain improved magnetic field measurements. R. S. Said, DWB & J. Twamley,

Physical Review B (accepted 19 January, 2011).

With low contrast, nonadaptive measurements are superior.

Anyon simulation Recall bosons and fermions

give different signs when exchanged.

Anyons are have more complicated behaviour – they give a phase or a more general group action.

Anyons can provide a basis for quantum computing with excellent error tolerance.

Simulated anyons can be produced experimentally.

Anyon simulation

Anyons are on a two-dimensional array of spins.

“Electric charges” are shown as diamonds.

“Magnetic charges” are shown as squares.

Charges correspond to excitations in the ground state of a Hamiltonian.

We take the smallest plaquette with nontrivial behaviour.

Anyon simulation

Method to produce the required state:

pump

single photon

10arcsin

247

7 3arcsin

42 26

DWB, M. Aguado, A. Gilchrist & G. K. Brennen, New Journal of Physics 12, 053011 (2010).

Photon processing

Two major problems for optical quantum information:

1. inefficiency of photon sources

2. photon loss

Can we recover from these problems using linear optics alone?

output

input

interferometer

measurement

…… ……

……

Photon processing Early results showed that we could

increase the single photon probability:

DWB, S. Scheel, B. C. Sanders & P. L. Knight, Physical Review A 69,

031806(R) (2004).

output

input

interferometer

measurement

…… ……

……

Photon processing Early results showed that we could

increase the single photon probability:

New results show that, once we have an appropriate definition of the efficiency, linear optics cannot increase the efficiency.

DWB, S. Scheel, B. C. Sanders & P. L. Knight, Physical Review A 69,

031806(R) (2004).

DWB & A. I. Lvovsky, Physical Review Letters 105, 203601 (2010).

output

input

interferometer

measurement

…… ……

……

Photon processing Early results showed that we could

increase the single photon probability:

New results show that, once we have an appropriate definition of the efficiency, linear optics cannot increase the efficiency.

Latest results indicate that we cannot use some high-efficiency sources to improve efficiency of other modes.

DWB, S. Scheel, B. C. Sanders & P. L. Knight, Physical Review A 69,

031806(R) (2004).

DWB & A. I. Lvovsky, arXiv:1010.6302 (2010).

DWB & A. I. Lvovsky, Physical Review Letters 105, 203601 (2010).

output

input

interferometer

measurement

…… ……

……

Photon processing A new way of quantifying vacuum in

modes.

Write annihilation operators for modes as

Vj are vacuum annihilation operators.

We form matrix of commutators

Non-vacuum component is quantified by Ky Fan k-norm of C.

j j ja B V

†[ , ]jn j nC B B

output

input

interferometer

measurement

…… ……

……

Quantum algorithms

Simulation of Hamiltonians

Quantum walks

Implementation of unitaries

Solving linear differential equations

Simulation of Hamiltonians Quantum computers could give an

exponential speedup in the simulation of quantum physical systems.

This is the original reason why Feynman proposed the idea of quantum computers.

The state of the system is encoded into the quantum computer.

Simulation of Hamiltonians The general problem is simulation of evolution under a

Hamiltonian.

This could be a quantum system – but a more general sparse Hamiltonian can encode some other problem!

/iHte

DWB, G. Ahokas, R. Cleve & B. C. Sanders, Comm. Math. Phys. 270, 359 (2007).

Simulation of Hamiltonians The general problem is simulation of evolution under a

Hamiltonian.

This could be a quantum system – but a more general sparse Hamiltonian can encode some other problem!

/iHte

NAND trees

A. M. Childs et al., Theory of

Computing 5, 119 (2009).

DWB, G. Ahokas, R. Cleve & B. C. Sanders, Comm. Math. Phys. 270, 359 (2007).

Simulation of Hamiltonians The general problem is simulation of evolution under a

Hamiltonian.

This could be a quantum system – but a more general sparse Hamiltonian can encode some other problem!

/iHte

NAND trees

A. M. Childs et al., Theory of

Computing 5, 119 (2009).

Systems of linear equations

A. W. Harrow et al., Phys. Rev. Lett. 103,

150502 (2009).

DWB, G. Ahokas, R. Cleve & B. C. Sanders, Comm. Math. Phys. 270, 359 (2007).

Simulation of Hamiltonians The general problem is simulation of evolution under a

Hamiltonian.

This could be a quantum system – but a more general sparse Hamiltonian can encode some other problem!

/iHte

NAND trees

A. M. Childs et al., Theory of

Computing 5, 119 (2009).

Systems of linear equations

A. W. Harrow et al., Phys. Rev. Lett. 103,

150502 (2009).

Differential equations

DWB, arXiv:1010.2745

(2010).

DWB, G. Ahokas, R. Cleve & B. C. Sanders, Comm. Math. Phys. 270, 359 (2007).

Quantum walks An entirely new approach to simulating Hamiltonians.

Quantum walks turn out to be universal for quantum computing!

A special type of quantum walk, called a Szegedy quantum walk, produces evolution related to that under the Hamiltonian.

By using a range of tricks, we can use the Szegedy quantum walk to simulate Hamiltonians far more efficiently.

wave

DWB & A. M. Childs, arXiv:0910.4157 (2009).

Implementation of unitaries A unitary is a general way of mapping a quantum state reversibly.

For dimension N, it takes at least N 2 elementary operations to perform the unitary (counting argument).

Alternatively, we can consider an oracle that gives the matrix elements of the unitary.

We can encode implementation of the unitary as a Hamiltonian simulation problem:

Then the complexity of performing the unitary, in most cases, scales as .

U

N

†

0

0

UH

U

DWB & A. M. Childs, arXiv:0910.4157 (2009).

Linear differential equations Most applications of supercomputers are in the form of large

systems of differential equations. A previous algorithm for nonlinear differential equations was not

efficient – try linear differential equations.

Using linear multistep methods, the problem can be encoded as solution of a linear system:

The complexity then scales as

Logarithmic in the dimension – an exponential speedup over classical solution.

Mx b

x Ax b

5/ 2logA t N

DWB, arXiv:1010.2745 (2010).

Research plansComplementing and enhancing the

research activities of the school.

Centre for Quantum Dynamics

ARC Centre for Quantum

Computation and

Communication Technology

Research plansComplementing and enhancing the

research activities of the school.

Centre for Quantum Dynamics

ARC Centre for Quantum

Computation and

Communication Technology

phase measurement

quantum algorithms

optical quantum

computing

Research plansComplementing and enhancing the

research activities of the school.

Centre for Quantum Dynamics

ARC Centre for Quantum

Computation and

Communication Technology

phase measurement

photon processing

anyon simulation

quantum algorithms

optical quantum

computing

Research plansPhase measurement

Primary challenge is to cope with photon loss.

1. Collaborate with Geoff Pryde & Centre for Quantum Dynamics to achieve experimental demonstration of proposal for loss tolerant states.

2. Develop new proposals for schemes with larger numbers of photons.

output

Research plansPhase measurement

Other collaborations:

1. Measurements of a fluctuating phase. Collaboration with Howard Wiseman (Centre for Quantum Dynamics) and researchers at UNSW and University of Tokyo to achieve adaptive measurements of a fluctuating phase with a squeezed beam.

2. Magnetometry with NV centres. Collaboration with Wrachtrup group at Universität Stuttgart and Jason Twamley at Macquarie University.

( )I t

Research plansOptical quantum computing

Primary challenge is again to cope with photon loss.

1. Parity states – methods to create and analyse. Collaboration with Geoff Pryde & Centre for Quantum Dynamics.

2. Develop new methods of optical quantum computing using ideas from simulation of nonabelian anyons.

3. Use photon processing theory to analyse loss tolerance in optical quantum computing.

pump

single photon

output

input

interferometer

measurement

…… ……

……

Research plansOptical quantum computing

Secondary challenge is to increase scale.

1. Hyperentanglement – exploit multiple degrees of freedom for each photon.

2. Heralded entanglement – enables more efficient construction of photonic cluster states.

3. Methods to use entangled particles to produce cluster states more directly. Possible collaboration with Dave Kielpinski.

Research plansQuantum algorithms

Solution of differential equations is an extremely promising area, with many open problems:

1. Can quantum walks be used for solving differential equations?

2. What information can be efficiently extracted from the states produced by algorithms for solving differential equations?

3. Can the efficiency be improved by using the variable time amplitude amplification of Ambainis?

4. Can time-dependent linear differential equations be efficiently simulated?

5. What about partial differential equations?

6. Are nonlinear differential equations fundamentally difficult to solve?

Summary

phase measurement

photon processing

anyon simulation

quantum algorithms

Summary

phase measurement

photon processing

anyon simulation

quantum algorithms

optical quantum

computing

Summary

Centre for Quantum Dynamics

ARC Centre for Quantum

Computation and

Communication Technology

phase measurement

photon processing

anyon simulation

quantum algorithms

optical quantum

computing

http://www.dominicberry.org/presentations/research.ppt